Example 3. A preliminary test is customarily given to the students at the beginning of a certain course. The following data are accumulated after several years: (a) 95% of the students pass the course, 5% fail. (b) 96% of the students who pass the course also passed the preliminary test. (c) 25% of the students who fail the course passed the preliminary test. What is the probability that a student who has failed the preliminary test will pass the course? Fail 95% preliminary test pass course 5% fail Figure 3.4 Let A be the event "fails preliminary test" and B be the event “Passes course." The probability we want is then PA(B) in (3.8), so we need P(AB) and P(A). P(AB) is the probability that the student both fails the preliminary test and passes the course; this is P(AB) = (0.95)(0.04) = 0.038. (See Figure 3.4; 95% of the students passed the course and of these 4% had failed the preliminary test.) We also want P(A), the probability that a students fails the preliminary test; this event corresponds to the shaded area in Figure 3.4. Thus P(A) is the sum of the probabilities of the two events "passes course after failing test," "fails course after failing test." Then P(A) = (0.095)(0.04) + (0.05)(0.75) = 0.0755 (See Figure 3.4; of the 95% of students who passed the course, 4% failed the prelimi- nary test; of the 5% of the students who failed the course, 75% failed the preliminary test since we are given that 25% passed.) By (3.8) we have P(AB) P(A) 0.038 PA(B) : 50%, 0.0755 that is, half of the students who fail the preliminary test succeed in passing the course. Note that in Figure 3.4, the shaded area corresponds to event A (fails preliminary test). We are interested in event B (passes course) given event A. Thus instead of the original sample space (whole rectangle in Figure 3.4) we consider a smaller sample space (shaded area in Figure 3.4). We then want to know what part of this sample space corresponds to event B (passes course). This fraction is P(AB)/P(A) which we computed.

Example 3. A preliminary test is customarily given to the students at the beginning of a certain course. The following data are accumulated after several years: (a) 95% of the students pass the course, 5% fail. (b) 96% of the students who pass the course also passed the preliminary test. (c) 25% of the students who fail the course passed the preliminary test. What is the probability that a student who has failed the preliminary test will pass the course? Fail 95% preliminary test pass course 5% fail Figure 3.4 Let A be the event "fails preliminary test" and B be the event “Passes course." The probability we want is then PA(B) in (3.8), so we need P(AB) and P(A). P(AB) is the probability that the student both fails the preliminary test and passes the course; this is P(AB) = (0.95)(0.04) = 0.038. (See Figure 3.4; 95% of the students passed the course and of these 4% had failed the preliminary test.) We also want P(A), the probability that a students fails the preliminary test; this event corresponds to the shaded area in Figure 3.4. Thus P(A) is the sum of the probabilities of the two events "passes course after failing test," "fails course after failing test." Then P(A) = (0.095)(0.04) + (0.05)(0.75) = 0.0755 (See Figure 3.4; of the 95% of students who passed the course, 4% failed the prelimi- nary test; of the 5% of the students who failed the course, 75% failed the preliminary test since we are given that 25% passed.) By (3.8) we have P(AB) P(A) 0.038 PA(B) : 50%, 0.0755 that is, half of the students who fail the preliminary test succeed in passing the course. Note that in Figure 3.4, the shaded area corresponds to event A (fails preliminary test). We are interested in event B (passes course) given event A. Thus instead of the original sample space (whole rectangle in Figure 3.4) we consider a smaller sample space (shaded area in Figure 3.4). We then want to know what part of this sample space corresponds to event B (passes course). This fraction is P(AB)/P(A) which we computed.

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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